The weather system will move southwards and by Thursday showers will become increasingly wintry.
A change in the wind by the start of Friday is set to bring snow showers to the North York Moors, as well as the north of Scotland, Northern Ireland, and maybe even Norfolk.
As the week progresses parts of the South East will also start to see wintry showers.
Steven Keates, from the Met Office, said: “On Thursday the showers spread further south and east, putting most places at risk of showers.
“It is going to turn much chillier, particularly through Thursday and Friday. Some areas could see snow, and some pretty significant snow.”
He added: “Most places are at risk. By Thursday and/or Friday, I think most places will stand risk of seeing something a bit wintry - sleet or snow.”
Temperatures are not set to dip too low this week, but the wind - and some risk of gales across Scotland - will make it feel colder than it is.
Mr Keates said: “Next weekend we could be into minus double figures overnight. But this week it will be more about the feel, rather than the temperatures.”